We will introduce stochasticity into the PDE solution. Below we will discuss how the probabilistic physics-informed neural networks [1]} can be used for UQ. When considering the stochasticity of a PDE, we assume the uncertainty mainly comes from two sources, the intrinsic uncertainty of the physics parameter \(\lambda \)  and the measurement uncertainty of the variable \(\mathbf {u}\) . In connection with this, we treat the physics parameter as a random variable, i.e. \(\lambda \sim p_{\lambda }(\lambda )\) , and assume the variable \(\mathbf {u}\)  follows a conditional distribution of the spatial-temporal coordinates, i.e. \(\mathbf {u} \sim p_{\mathbf {u}}(\mathbf {u}|x,t)\) .
